?
Координатное и импульсное туннелирование в одномерных квантовых системах с дискретным спектром
Наноструктуры. Математическая физика и моделирование. 2015. Т. 12. № 1. С. 5-84.
Vybornyi E.
We consider the problem of constructing semiclassical asymptotic expansions of discrete spectrum and the corresponding stationary states of one-dimensional Schrödinger operator in the case of resonance tunneling. We consider two basic models: tunneling in an asymmetric double-well potential on a line and momentum tunneling of a particle in a potential field on a circle. For an asymmetric double-well potential we obtain the criterion of resonance tunneling, i. e. the necessary and sufficient conditions of stationary states bilocalization. We obtain explicit asymptotic formulas for the tunneling energy splitting in the case of high energy levels and for energies close to the minima of the potential. In the general case of dynamic tunneling we proposed a general method to find the asymptotic estimates for the tunneling energy splitting. In the case of the particle on a circle our method yields an asymptotic formula for the tunneling splitting, which is applicable in the case of analytical potential as well as in the case of finite smoothness. As an example, we consider the problem of momentum tunneling of the quantum pendulum.
Vybornyi E., Наноструктуры. Математическая физика и моделирование 2015 Т. 13 № 2 С. 43-54
We conceder semiclassical asymptotics of the energy levels shift of the Schrödinger operator discrete spectrum with a one-dimensional single-well potential that appears due to a deformation of the potential in the classically forbidden region. Since such a deformation of the potential effects on the quantum particle only due to the tunneling effects, then the corresponding ...
Added: February 18, 2016
Karasev M., Vybornyi E., / Cornell University. Series "Working papers by Cornell University". 2014. No. 1411.4436.
We consider the one-dimensional Schr\"{o}dinger operator in the semiclassical regime assuming that its double-well potential is the sum of a finite "physically given" well and a square shape probing well whose width or depth can be varied (tuned). We study the dynamics of initial state localized in the physical well. It is shown that if ...
Added: November 18, 2014
Vybornyi E., Theoretical and Mathematical Physics 2014 Vol. 181 No. 2 P. 1418-1427
We propose an operator method for calculating the semiclassical asymptotic form of the energy splitting value in the general case of tunneling between symmetric orbits in the phase space. We use this approach in the case of a particle on a circle to obtain the asymptotic form of the energy tunneling splitting related to the ...
Added: August 5, 2014
Vybornyi E., Theoretical and Mathematical Physics 2014 Vol. 178 No. 1 P. 93-114
We consider the one-dimensional stationary Schr¨odinger equation with a smooth double-well potential. We obtain a criterion for the double localization of wave functions, exponential splitting of energy levels, and the tunneling transport of a particle in an asymmetric potential and also obtain asymptotic formulas for the energy splitting that generalize the well-known formulas to the ...
Added: December 23, 2013
Karasev M., Vybornyi E., Journal of Mathematical Physics 2016
We consider the one-dimensional Schrodinger operator in the semiclassical regime assuming that its double-well potential is the sum of a finite "physically given" well and a square shape probing well whose width or depth can be varied (tuned). We study the dynamics of an initial state localized in the physical well. It is shown that ...
Added: October 23, 2015
Vybornyi E., В кн. : Научно-техническая конференция студентов, аспирантов и молодых специалистов МИЭМ НИУ ВШЭ. Тезисы докладов. : М. : МИЭМ НИУ ВШЭ, 2013. С. 10-10.
Рассмотрено влияние туннельного возмущения на дискретный спектр оператора Шредингера. Возмущение потенциала является финитной носитель, которой не пересекается с областью классического движения частицы. ...
Added: March 5, 2013
Anikin A. Y., Brüning J., Dobrokhotov S. et al., Russian Journal of Mathematical Physics 2019 Vol. 26 No. 3 P. 265-276
In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can ...
Added: September 18, 2019
Pereskokov A., Липская А. В., Вестник Московского энергетического института 2010 № 6 С. 99-109
Рассмотрены радиально-симметричные решения уравнения типа Хартри, содержащего как кулоновский потенциал, так и интегральную нелинейность с потенциалом взаимодействия Юкавы. В квазиклассическом приближении выведены и исследованы уравнения для самосогласованного потенциала. Выписано правило квантования типа Бора-Зоммерфельда. Найдены асимптотические собственные значения и собственные функции. ...
Added: December 16, 2012
Karasev M., Russian Journal of Mathematical Physics 2012 Vol. 19 No. 3 P. 299-306
For the Dirac 2D-operator in a constant magnetic field with perturbing electric potential, we derive Hamiltonians describing the quantum quasiparticles (Larmor vortices) at Landau levels. The discrete spectrum of this Hall-effect quantum Hamiltonian can be computed to all orders of the semiclassical approximation by a deformed Planck-type quantization condition on the 2D-plane; the standard magnetic (symplectic) ...
Added: December 19, 2012
Chernyshev V.L., Russian Journal of Mathematical Physics 2016 Vol. 23 No. 3 P. 348-354
On a two-dimensional surface, a Schrödinger operator is considered with a potential whose critical points form a closed curve. We pose the problem of describing the semiclassical spectral series corresponding to this curve. The standard construction for describing the spectral series corresponding to isolated nondegenerate equilibria or to periodic trajectories of Hamiltonian systems is not ...
Added: October 25, 2014
Bruning J., Grushin V. V., Dobrokhotov S. Y., Математические заметки 2012 Т. 92 № 2 С. 163-180
An example of Schrodinger and Klein-Gordon equations with fast oscillating coefficients is used to show that they can be averaged by an adiabatic approximation based on V.P. Maslov's operator method. ...
Added: December 24, 2012
Vybornyi E., Karasev M., Наноструктуры. Математическая физика и моделирование 2014 Т. 11 № 1 С. 27-36
We consider the one-dimensional Schrodinger operator in the semiclassical regime assuming that its double-well
potential is the sum of a finite “physically given” well and a square shape probing well whose width or depth can
be varied (tuned). We study the dynamics of initial state localized in the physical well. It is shown that if the probing
well ...
Added: October 23, 2014
Karasev M., Novikova E., Vybornyi E., Mathematical notes 2016 Vol. 100 No. 6 P. 807-819
We describe how a top-like quantum Hamiltonian over a non-Lie algebra appears in the model of the planar Penning trap under breaking its axial symmetry (inclination of the magnetic field) and turning parameters (electric voltage, magnetic field strength and inclination angle) at double resonance. For eigenvalues of the quantum non-Lie top, under a specific variation ...
Added: October 22, 2016
Ducomet Bernard, Zlotnik Alexander, Zlotnik Ilya, ESAIM: Mathematical Modelling and Numerical Analysis 2014 Vol. 48 No. 6 P. 1681-1699
We consider an initial-boundary value problem for a generalized 2D time-dependent Schrödinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time $L^2$-stability is proved. ...
Added: May 23, 2014
Arseyev P., Маслова Н. С., The European Physical Journal B 2015 Vol. 88 No. 2 P. 40:1-40:9
We proved that for arbitrary mixed state the concurrence and the entanglement are determined by the average value of electron’s pair correlation functions particular combinations. We analyzed the dynamics of the initial two-electronic state in two interacting single-level quantum dots (QDs) with Coulomb correlations, weakly tunnel coupled with an electronic reservoir. We obtained correlation functions ...
Added: March 17, 2016
Lasukov V. V., Abdrashitova M.O., Russian Physics Journal 2020 Vol. 63 No. 4 P. 631-648
A quantum solution of the classical electrodynamics equations has been found. It is shown that all information on the multiparticle process of creation of scalar pairs of particles by a nonstationary self-acting electric field is contained in solutions of the d’Alembert single-particle equation. The existence of a quantum solution of the d’Alembert equation is determined ...
Added: September 7, 2020
Remizov I., Starodubtseva M., Математические заметки 2018 Т. 104 № 5 С. 790-795
краткое сообщение - нет аннотации ...
Added: September 29, 2018
Vybornyi E., Karasev M., Russian Journal of Mathematical Physics 2018 Vol. 25 No. 4 P. 500-508
We consider a charge in a general electromagnetic trap near a hyperbolic stationary point. The two-dimensional trap Hamiltonian is a sum of hyperbolic harmonic part and higher order anharmonic corrections. We suppose that two frequencies of the harmonic part are under a resonance 1 : (-1). In this case, anharmonic terms define the dynamics and ...
Added: November 16, 2018
Karasev M., Russian Journal of Mathematical Physics 2016 Vol. 23 No. 4 P. 483-489
We discuss two examples of classical mechanical systems which can become quantum either because of degeneracy of an integral of motion or because of tuning parameters at resonance. In both examples, the commutativity of the symmetry algebra is breaking, and noncommutative symmetries arise. Over the new noncommutative algebra, the system can reveal its quantum behavior ...
Added: October 22, 2016
Arseyev P., Mantsevich V. N., Maslova N. S., JETP Letters 2012 Vol. 95 No. 10 P. 589-594
The possibility of non-adiabatic electron pumping in the system of three coupled quantum dots attached to the leads is discussed. We have found out that periodical changing of energy level position in the middle quantum dot results in non zero mean tunneling current appeared due to non-adiabatic non-equilibrium processes. The same principle can be used ...
Added: October 28, 2014
Zlotnik A., Zlotnik I. A., Kinetic and Related Models 2012 Vol. 5 No. 3 P. 639-667
We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large x. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in ...
Added: March 21, 2013
Zlotnik A., Romanova A. V., / Cornell University. Series math "arxiv.org". 2013. No. arxiv: 1307.5398.
We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semi-infinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L_2-stability (in particular, L_2-conservativeness) are ...
Added: July 24, 2013
Novikova E., Наноструктуры. Математическая физика и моделирование 2012 Т. 7 № 2 С. 59-86
Рассматривается спектральная задача для атома водорода, помещенного в возмущающие магнитное и электрическое поля. Найдены резонансные соотношения на величину полей и угол между ними, при которых квантовая усредненная система (в первом порядке теории возмущений) имеет нелиевскую алгебру симметрий, чьи коммутационные соотношения между генераторами задаются полиномами не выше кубического. Неприводимые представления этой алгебры соответствуют спектральным кластерам, локализованным вблизи ...
Added: December 22, 2012
Galkin O., Galkina S., / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2012.07174.
This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their properties. The results are given without proofs but the proofs will appear in the longer text which is now under preparation.
In ...
Added: December 13, 2020